Raman spectra of SDW superconductors

Physica C 420 (2005) 37–50 www.elsevier.com/locate/physc

Raman spectra of SDW superconductors G.C. Rout a

a,*

, K.C. Bishoyi b, S.N. Behera

c,1

Condensed Matter Physics Group, Department of Physics, Government Science College, Chatrapur, Orissa 761 020, India b P.G. Department of Physics, F.M. College (Autonomous), Balasore, Orissa 756 001, India c Institute of Physics, Bhubaneswar 751 005, India Received 21 September 2004; received in revised form 22 December 2004; accepted 13 January 2005

Abstract We report the calculation of the phonon response of the coexistent spin density wave (SDW) and superconducting (SC) state and predict the observation of SC gap in the Raman spectra of rare-earth nickel borocarbide superconductors. The SDW state normally does not couple to the lattice and hence, the phonons in the system are not expected to be affected by the SDW state. But there is a possibility of observing SC gap mode in the Raman spectra of a SDW superconductor due to the coupling of the SC gap excitation to the Raman active phonons in the system via the electron– phonon (e–p) interaction. A theoretical model is used for the coexistent phase and electron–phonon interaction. Phonon Green
s function is calculated by Zubarev
s technique and the phonon self-energy due to e–p interaction which is given by electron density response function in the coexistent state corresponding to the SDW wave vector q = Q is evaluated. The results so obtained exhibit agreement with the experimental observations.
2005 Elsevier B.V. All rights reserved. PACS: 74.72.Ny; 74.72.Y Keywords: Quaternary borocarbides; Superconductivity of borocarbides

1. Introduction A large family of quaternary borocarbides RNi2B2C (R = Lu, Y, Ho, Dy) was discovered *

Corresponding author. Fax: +91 674 2300142. E-mail address: [email protected] (G.C. Rout). 1 Present address: HIG-23/1, Housing Board Phase-1, Chandrasekharpur, Bhubaneswar 751 016, India.

which exhibit separate phases of antiferromagnetism (AFM), superconductivity (SC) as well as coexistent phase of antiferromagnetism and superconductivity. These superconductors were first discovered in India [1]. A brief review of these borocarbide systems can be found in the papers [2–5]. The non-magnetic compounds such as YNi2B2C and LuNi2B2C have rather high value of Tc with transition temperature of 15.5 K and

0921-4534/$ - see front matter
2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2005.01.006

38

G.C. Rout et al. / Physica C 420 (2005) 37–50

16.6 K respectively. These are probably the first to exhibit superconductivity with a sizable amount of nickel in them. The compounds like RNi2B2C (R = Tm, Er, Ho, Dy) exhibit coexistence of AFM and SC long range orders. Among these, there are systems with Ne´el temperature (TN) higher than Tc as in DyNi2B2C with TN = 10 K and Tc = 6 K and others with Tc > TN, with highest ratio of Tc/TN being 11/1.5 ’ 7.33 for TmNi2B2C. The compounds RNi2B2C (R = Tb, Gd) are pure antiferromagnets with highest value of TN being 20 K for GdNi2B2C. The Ne´el temperatures (TN) of these quaternary compounds show a systematic almost linear increase with the increasing atomic number as well as the de-Gennes factors of the rare-earth atoms. Simultaneously the SC transition temperature in these systems show a systematic suppression [5]. To obtain some insight into the subtle interplay between superconductivity and magnetism, the magnetic structures of the rare-earth nickel borocarbide (RNi2B2C; R = rare-earth) have been studied extensively by neutron and X-ray scattering [6,7]. The superconducting Er [8,9] and Ho [10,11] compounds as well as non-superconducting Tb [12] and Gd [13] compounds show evidence for the existence of order in an incommensurate magnetic structure characterised by a wave vector Qm (0.55, 0, 0) for all of these compounds. The band structure calculations [14] of the generalised electron susceptibility v(q) of the LuNi2B2C show Fermi surface nesting, characterised by a common nesting wave vector Qm
for these compounds. The existence of such Fermi surface nesting suggests that strong Kohn anomalies should be observed in the phonon dispersion curves of these compounds for phonons with wave vector close to Qm. In addition, electronic band structure calculations [15–18] suggest that these materials are conventional superconductors arising due to strong coupling of the phonons to the electrons in a complex set of bands crossing at the Fermi level (
F). The detailed study of the low lying excitations in the compounds of Lu and Y based borocarbides showed that there is a strong interaction between the phonons of the acoustic branch and the low lying optical branch above the superconducting transition temperature [19–21]. A

sharp peak was observed at 4.5 meV with a weak shoulder at the higher energy side at temperatures well below Tc. The width and position of the sharp peak remain practically unchanged below Tc and its intensity decreases with the increase in temperature on approaching Tc. These features are suggested by the authors [20,21] to be due to the interplay of SC, SDW and their coupling to the phonon mode. Hence, it is essential to study the phonon response of the zone boundary phonons corresponding to the nesting wave vector (Q) in the coexistent phase of SDW and SC states. Similarly the high-Tc superconducting cuprate materials also exhibit the antiferromagnetic phase for low dopant concentrations with indirect evidence for the coexistence of the AFM state with the SC state. High-Tc perovskite superconductors La2
xSrxCuO4 and YBa2Cu3O7
d which have a two dimensional character are also likely candidates for the coexistence of SDW and SC [22]. In these materials, some of the optic phonons observed in the Raman spectra show large shifts in their frequency on going to the SC state [23]. Schrieffer et al. [24] have proposed the presence of the spin density wave (SDW) state as one of the reasons for the enhancement of Tc. The SDW and the charge density wave (CDW) states arise from the nesting property of the Fermi surface in low dimensional systems. The SDW state arises due to Coulomb interaction between the electrons while the CDW state is a consequence of the electron–phonon interaction in presence of perfectly nested pieces of Fermi surface in low dimensional systems. It is well known that in some layered compounds due to the coexistence of CDW and SC, the SC gap mode becomes observable in Raman spectrum by borrowing Raman activity from the CDW amplitude mode [25,26]. Hence, we attempt in this work to calculate the phonon response of the system with a view to examine the possibility of observing the SC gap mode in presence of the SDW state by Raman scattering. However, SDW is not normally coupled to the lattice and hence the phonons in the system are not expected to be effected by the SDW state. Since Raman scattering usually proceeds by exciting the zone centre (q = 0 and q = Q, the later due to zone folding) optical phonon, the SDW amplitude

G.C. Rout et al. / Physica C 420 (2005) 37–50

mode may not be seen in the Raman spectrum. But, there is still the possibility of observing SC gap mode in Raman spectra of a SDW–SC, due to the coupling of the SC gap excitations to the Raman active phonons in the system via the electron–phonon interaction. Hence, attempt is made here to calculate the phonon response which is proportional to the Raman intensity in the SDW and SC states, to observe SC gap in the Raman spectra. For the coexistence of SDW and SC, it is required that the phonon self-energy be calculated in the coexistent state, rather than in the pure SC state. The rest of this paper is organised as follows. The theoretical model for the SDW and SC coexistent phase and the electron–phonon interaction is formulated in Section 2. The phonon Green
s function is calculated using the Zubarev
s technique of equations of motion method in Section 3. The phonon self-energy due to the electron– phonon interaction which is given by electron density response for the coexistent SDW and SC state for the nesting wave vector q = Q is evaluated in Section 4. Finally the results and discussion are presented in Section 5 with reference to the borocarbide and high-Tc superconductors. The conclusion is given in Section 6.

2. Theoretical model In high-Tc material, some of the optic phonons observed in the Raman spectra [23] show large shifts in their frequency on going to the SC state. This indicates a strong electron–phonon coupling in these systems. Hence, there have been attempts [27] to analyse these shifts using the strong coupling theory. Since Tc enhancement is proposed to take place within a model [28] in presence of the SDW state, the phonon self-energy calculation in the present work is aimed at looking at similar effects in the coexistent phase of the system. A mean-field Hamiltonian is constructed for the SDW state starting from the Hubbard Hamiltonian. Similarly, the SC phase is described by the BCS reduced Hamiltonian, where, the effective attractive interaction between the electrons is produced by the suitable exchange of an excitation,

39

which may or may not be the phonon. In particular, in the presence of the SDW state, this excitation could be the SDW amplitude mode, as argued by Schrieffer et al. [28] the exchange of which can give rise to an attractive interaction between two electrons as well as to a high transition temperature. The mean-field Hamiltonian describing the coexistent SDW–SC state is given by X X y
k C yk;r C k;r þ G ðC kþQ" C k# þ C yk# C kþQ" Þ H0 ¼ k;r

þD

X

k

ðC yk" C y
k#

þ C
k# C k" Þ

ð1Þ

k

where C kr ðC ykr Þ are the annihilation(creation) operators for the conduction electrons derived from the Nickel atom with momentum k and spin r. The G and D are respectively the SDW and SC order parameters given by X y G ¼
U hC kþQ" C k# i ð2Þ k

D ¼
V

X

hC yk" C y
k# i

ð3Þ

k

with U and V being respectively the repulsive Coulomb and attractive interaction strengths, and Q being the nesting wave vector. In defining the superconducting order parameter (D) s-wave singlet pairing is assumed as per the usual BCS prescription. In Eq. (1) the transverse polarisation of the SDW is considered and the nesting of the pieces of the Fermi surface separated by the wave vector Q responsible for the formation of the SDW results in the electron–hole symmetry.
kQ ¼

k

ð4Þ

In what follows complete nesting over the entire Fermi surface is assumed. The Hamiltonian in Eq. (1) can be written in generalised Nambu formalism as given below [29]. X y H¼ wk ð
s q0 r3 þ
a q3 r3 þ q3 r1 D þ q1 r3 GÞwk k

ð5Þ where
s;a ¼ ð
k 
kþQ Þ=2

ð6Þ

40

G.C. Rout et al. / Physica C 420 (2005) 37–50

and the four-component field operators are defined as wyk

¼

ðC yk"

C
k# C ykþQ#

C
k
Q" Þ

ð7Þ

Further the (4 · 4) matrices ri and qi are defined as
 si 0 ri ¼ 0 si with si being the (2 · 2) Pauli matrices, and

 0 I 0
iI q1 ¼ ; q2 ¼ ; I 0 iI 0
 I 0 q3 ¼ 0
I

ð8Þ

k;q

where the qth-Fourier transformation of the phonon displacement operator is ð10Þ

and f being the electron–phonon coupling constant, bq ðbyq Þ are phonon annihilation(creation) operators with wave vector q and the free phonon Hamiltonian (Hp) is given by X Hp ¼ xq byq bq ð11Þ q

where xq is the free phonon frequency. Combining Eqs. (5), (9) and (11), the total Hamiltonian of the system is given by H ¼ H 0 þ H e–p þ H p

¼
iHðt
t0 Þh½Aq ðtÞ; Aq0 ðt0 Þi

ð13Þ

Using the Hamiltonian of Eq. (12), the Fourier transformed phonon Green
s function is found to be Dq;q0 ðxÞ ¼ d
qq0 D0q ðxÞ þ p2 f 2 D0q ðxÞvðq; xÞD0q0 ðxÞ ð14Þ where the free phonon propagator is given by xq 2
1 D0q ðxÞ ¼ ðx
x2q Þ ð15Þ p

Here I is the (2 · 2) unit matrix and q0 and r0 are the (4 · 4) unit matrices. In order to calculate the phonon response in the borocarbide superconductors, one has to take into account electron–phonon interaction in the system. Here coupling of the phonon to the conduction electrons is given by X y H e–p ¼ f wkþq q0 r3 wk Aq ð9Þ

Aq ¼ bq þ by
q

Dq;q0 ðt
t0 Þ ¼ hhAq ðtÞ; Aq0 ðt0 Þii

ð12Þ

3. Calculation of phonon Green
s function In order to evaluate the phonon response of the system, one has to calculate the phonon Green
s function defined as

v(q, x) being the electron density response function. Applying Dyson
s approximation to Eq. (14), the phonon Green
s function can be written as xq 2 Dqq ðxÞ ¼ ½x
x2q
Rðq; xÞ
1 ð16Þ p where phonon self-energy is given by R(q, x) = pxqf 2v(q, x) and the electron density response function v(q, x) is defined by vðq; xÞ X y hhwkþq ðtÞq0 r3 wk ðtÞ; wyk0
q ðt0 Þq0 r3 w0k ðt0 Þiix ¼ k;k 0

ð17Þ It is evident from Eq. (16) that the phonon selfenergy involves the electron response of the coexistent SDW–SC state and v(q, x) is evaluated using the mean-field Hamiltonian (H0) by equations of motion method. The calculation of v(q, x) involves four sets of coupled matrix equations. In addition to this, there arises a great deal of simplification in the solution of these coupled equations due to the nesting property given in Eq. (4). In the present case, the presence of the SDW state produces a super-lattice, with periodicity equal to the wave length k = 2p/Q of the spin density wave in the direction of Q. As a consequence Q becomes the new reciprocal lattice vector, due to which, the phonons with wave vector Q will also show up in the zone centre. Hence, we calculate v(q, x) for q = Q corresponding to the optic phonons in the system arising from zone folding in the presence of SDW state. For q = Q, the response function reduces to

G.C. Rout et al. / Physica C 420 (2005) 37–50

vðQ; xÞ ¼

2 XX k

i¼1

4 pEik jD0 ðk; xÞj

 ½ðx2
4G2 ÞDDi þ x2
2k  tanh Z

þW2


 bEik 2

2 X

4 pE jD ik 0 ðk; xÞj i¼1
 bEik  ½ðx2
4G2 ÞDDi þ x2
2k  tanh 2

¼ N ð0Þ

d
k


W2

ð18Þ where in the second step in Eq. (18), the summation over k is replaced by an integral over
k assuming the density of states at the Fermi level to be a constant N(0) and the different quantities entering Eq. (18) are defined as jD0 ðk; xÞj ¼ ½x2 ðx2
4E2k Þ þ 16G2 D2 

ð19Þ

D1;2 ¼ ðD  GÞ

ð20Þ 1=2

Eik ¼ ð
2k þ D2i Þ

;

Ek ¼ ð
2k þ G2 þ D2 Þ

1=2

ð21Þ In Eq. (20), the SDW and SC order parameters G and D appear as a sum and difference indicating the possible interference between these two quantities [29], which is expected because these are the expectation values of " and # spin electron–hole and two electron pairs in momentum space. In real space, these being simply the wave functions of the corresponding pairs, it is natural that each of these wave functions will have also their own phase factors. In quantities where the modulus squares of G and D appear as in Ek in Eq. (21), the phase factors will not count. But in quantities like Eik, where the squares of Di (i = 1, 2) appear, the relative phase can play an important role . However in what follows we do not take into account the relative phase factor explicitly. For the SDW phonon i.e. the phonon with wave vector Q , the response function is non-zero for D = 0 as well as G = 0 , as can be seen from Eq. (18). For pure SDW state (D = 0), Eq. (18) reduces to  1 bE X
2k tanh 2 k vðQ; xÞ ¼ 8 ð22Þ 1 12 2 k 2pEk ðx
4Ek Þ

41

where the excitations of the SDW state are E1k ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2k þ G2 . Similarly for pure superconducting state (G = 0)  2 bE X E2k tanh 2 k vðQ; xÞ ¼ 8 ð23Þ 22 2 k 2pðx
4Ek Þ where the excitation energy of the SC state is E2k ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2k þ D2 . From Eqs. (22) and (23), it is obvious that the response function is non-zero for D = 0 as well as G = 0 for the SDW phonons (the phonons with wave vector Q) unlike the case of the zone centre optic phonon corresponding to q = 0 in the absence of the SDW state. Hence the phonon self-energy in SDW–SC state gets modified.

4. Expression for the phonon spectral density function S(Q, x) The phonon spectral density function (SDF) being the imaginary part of the phonon Green
s function, its evaluation essentially boils down to the calculation of v(q, x) (see Eq. (17)), which in turn is evaluated using the mean-field Hamiltonian H0 given in Eq. (1) for the SDW–SC state. The phonon spectral density function for the zone folded phonon (q = Q, optic phonon) is defined through the equation S(x, Q) =
2Im Æ DQQ(x + ig) with (x2 ! x2 + 2ixg), g being the small but finite width given to the phonon frequency. The phonon spectral density function thus calculated is given by, SðQ; xÞ ¼

x2Q B1 ðxÞ

ð24Þ

p½A21 ðxÞ þ B21 ðxÞ

where A1 ðxÞ ¼ ½x2
g2
x2Q ð1 þ A2 Þ; B1 ¼ ½2gx
x2Q B2  with A2 and B2 expressed as Z W =2 Z d
k Gð
k Þ; B2 ¼ 4s A2 ¼ 4s
W =2

W =2

d
k H ð
k Þ


W =2

ð25Þ

42

G.C. Rout et al. / Physica C 420 (2005) 37–50

where s = N(0)f 2(Q)/xQ is the dimensionless coupling constant involving the strength of the electron–phonon interaction (f ) while the other quantities are defined as 1 þ Q2  1

 P 31 bE1k P 32 bE2k  tanh tanh þ E1k 2 E2k 2 1 H ð
k Þ ¼ 2 P 1 þ Q21 

 Q bE1k Q bE2k  31 tanh þ 32 tanh E1k 2 E2k 2 Gð
k Þ ¼

P 21

ð26Þ P 31 ¼ P 1 P 21 þ Q1 Q21 ; Q31 ¼ P 1 Q21
Q1 P 21 ; P 32 ¼ P 1 P 22 þ Q1 Q22 ; Q32 ¼ P 1 Q22
Q1 P 22 ; P 21 ¼ ðx2
4G2
g2 ÞDD1 þ ðx2
g2 Þ
2k ; Q21 ¼ 2gxðDD1 þ
2k Þ; P 22 ¼ ðx2
4G2
g2 ÞDD2 þ ðx2
g2 Þ
2k ; Q22 ¼ 2gxðDD2 þ
2k Þ; P 1 ¼ ðx2
x21
g2 Þðx2
x22
g2 Þ
4g2 x2 ; Q1 ¼ 2gxð2x2
2g2
x21
x22 Þ; x21 ¼ 2½E2k þ E1k E2k ; x22 ¼ 2½E2k
E1k E2k  ð27Þ with D1,2, Ek and Eik as defined in Eqs. (20) and (21). Further more, the cut off energy Es of the SDW state is defined as the energy above the Fermi energy which destroys the nesting of the Fermi surface completely. Similarly xD is the highest frequency of the boson whose exchange brings about the attractive interaction between the electrons. All the calculations are carried out in terms of the quantities entering Eq. (24) which are made dimensionless by dividing those with xQ. Thus the dimensionless order parameters are redefined as the SC order parameter, z = D/xQ; the SDW order parameter, g = G/xQ; temperature t = kBT/xQ; the conduction band width, W = W/xQ; the width

of the phonon, e = g/xQ; the SDW phonon at the zone centre being, p = xQ/xQ = 1 and the variable ~ ¼ x=xQ . reduced frequency, x

5. Results and discussion The evaluation of the spectral density function (SDF) requires the judicious choice of specific values of the parameters such as the strength of the electron–phonon coupling (s), the position (p) and width (e) of the zone folded SDW phonon with q = Q, as well as the values of other quantities entering Eq. (24) such as the SC order parameter (z), the SDW order parameter (g) and the temperature (t). The later quantities such as z and g can be explicitly evaluated at any temperature t by self-consistently solving the corresponding coupled gap equations as has been done in [29,30]. However, in what follows the values of these quantities are chosen in order to accentuate the desired effect namely the observation of the peaks corresponding to the SC and SDW gaps in the SDF. Since all the energies are normalised with respect to the SDW phonon frequency xQ, automatically the value of p = 1 in the absence of interaction. The value of the width parameter is taken to be e = 0.025 for the evaluation of SDF in all the cases examined. The electron–phonon interaction strength s is approximately chosen to show the desired effect in the SDF. The value of the superconducting order parameter (z) is chosen such that the BCS relation 2D(0)/kBTc = 3.5 is satisfied approximately. This requirement is satisfied for the choice z(0) = 0.176, which automatically fixes the value of SC transition temperature to be tc = 0.1. Thus choosing the value of z to be same as z(0) further requires the temperature should be kept close to T = 0 K, which is fulfilled by taking t = 0.001. In contrast the SDW order parameter is chosen to be very small g(0) = 0.0352, which according to the mean-field theory requires the Ne´el temperature to be tN = 0.0201, an order of magnitude smaller than the SC transition temperature tc. Thus the above choice of parameters corresponds to systems with tc > tN in which the two long range orders coexist below tN. Furthermore for this choice of parameters, the SDW phonon couples

G.C. Rout et al. / Physica C 420 (2005) 37–50

to the SC gap mode only above a threshold value of the electron–phonon coupling constant (s) which happens to be s = 0.035. The gap mode cannot be seen for s < 0.035. The phonon spectral density function as a function of reduced frequency is plotted in Fig. 1, for the cases where either the SC state or SDW state only exists. For a weak electron–phonon coupling (in the range s = 0.035–0.040), only one phonon excita~ ¼ 0:335 along with a tion peak (P1) appears at x ~ ¼ 0:9. The peak P1 bare phonon peak (P0) at x is the superconducting gap excitation peak appear~ ’ 2z as can be seen from Fig. 1. No phoing at x non excitation peak is observed for s > 0.035. This value of s is much smaller in comparision to the choice of s = 0.11–0.16 for the charge density wave (CDW) superconductors [26,31] with the zone centre phonons (q = 0) coupling to the SC gap excitation. Similar choice of a small value of s = 0.025 has also been made for the high-Tc superconductors in the normal phase with AFM ground state. The figure also demonstrates the fact that the zone folded phonons (q = Q) cannot excite SDW gap excitation peak in this weak coupling limit in the absence of superconductivity. This is clearly depicted by the dotted curve in the inset of Fig. 1. This is so because, the phonon is not directly coupled to the SDW order parameter. However,

43

a very small flattened peak can be seen near ~ ¼ 0:07. The emergence of this peak may be x due to the interplay of the SDW order parameter and the weak electron–phonon coupling. This fact is further elaborated in the discussion of Fig. 3. The bare phonon peak (P0) gets shifted to a lower ~ ¼ 0:9 due to the electron–phonon frequency x interaction. In order to probe whether the flattened peak in the inset of Fig. 1 really corresponds to the SDW ~ plot, we increased the elecpeak in the SDF vs. x tron–phonon coupling strength from the weaker limit to a stronger values (i.e. from 0.04 and 0.06) in the absence of SC (i.e., with z = 0). Fig. ~ 2 ¼ 0:07 (see Fig. 2). This 2 shows a peak (P2) at x new peak which shows up on increasing the coupling constant appears at the energy of the SDW gap  2g. This peak is further explored in Fig. 3 whose frequency and width increases with increasing value of the order parameter g. It can also be seen that the strong electron–phonon coupling tends to distort the bare phonon peak (P0). The later effects may be an outcome of an incipient lattice instability arising from the assumed perfect nesting and a strong electron–phonon interaction. Under these conditions, the CDW state may be competing with SDW state. As a result, the spectral weight of (P2) may be increasing with a shift of its position towards lower

30

z=0.176, g=0, s=0.04 z=0.000, g=0.0352

0.3

100

20

P2

P0

0.1 0

0

0.1

0.2

0.3

0.4

s=0.06 , z=0 , g=0.0352

0.5

SDF

spectral density function (SDF)

P1 0.2

P1

50

10

P0

0

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

reduced frequency

Fig. 1. The plot of spectral density function S(Q, x) vs. reduced ~ ¼ ðx=xQ Þ for fixed values of e = 0.025, p = 1.0, frequency x t = 0.001, s = 0.04 with values of z and g being given in the plot.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

reduced frequency

Fig. 2. The plot of spectral density function S(Q, x) vs. reduced ~ ¼ ðx=xQ Þ for fixed values of e = 0.025, p = 1.0, frequency x t = 0.001, s = 0.06, z = 0, g = 0.0352.

44

G.C. Rout et al. / Physica C 420 (2005) 37–50 150 100

100

100

P2

P2

P2 50

50

SDF

0 0.04

0.06

SDF

P2

100

0.08

0 0.02

50

s=0.060, z=0, g=0.0352 g=0.0362 g=0.0372

50

0.04

0.06

0.08

0.1

t=0.001, s=0.06 , z=0, g=0.0352 t=0.009 t=0.012

P0 P0

0

0 0

0.25

0.5

0.75

1

0

0.25

reduced frequency

energy on decreasing the SDW order parameter (g). This is clearly shown in the inset of Fig. 3. Hence, it is possible that the SDW gap parameter can be experimentally detected through the spectral density function in presence of a strong electron–phonon interaction. Moreover, a strong SDW order parameter is expected to be accompanied by a weak electron–phonon interaction as observed by the low value of the isotope exponent (a ’ 0.2) [32,33] in certain superconductors. The temperature dependence of the peak (P2) of Fig. 3 is shown in Fig. 4. It is observed that the spectral weight of the peak (P2) decreases with increase of temperature. There is a slight shift in the position of the peak towards higher frequencies. It should be noted that with increasing temperature, g is also expected to decrease which is not taken into account in Fig. 4. The peak (P2) in Fig. 3 appears only due to strong electron–phonon interaction (s = 0.060) in the SDW phase; in the absence of the superconducting phase. When the electron–phonon interaction is increased further from s = 0.060 to s = 0.063, the peak (P2) splits into two peak P2a ~ ’ 0:04 and P2b at x ~ ’ 0:083 as shown in at x Fig. 5. The peak P2a may be a consequence of excitation arising from the distortion caused by the

0.75

1

Fig. 4. The plot of spectral density function S(Q, x) vs. reduced ~ ¼ ðx=xQ Þ for fixed values of e = 0.025, p = 1.0, frequency x s = 0.06, z = 0, g = 0.0352 and different values of the reduced temperature t = 0.001, 0.009, 0.012.

400 100

P 2a

P 2a

80

P2

P 2b

60

300

40

P 2b

SDF

Fig. 3. The plot of spectral density function S(Q, x) vs. reduced ~ ¼ ðx=xQ Þ for fixed values of e = 0.025, p = 1.0, frequency x t = 0.001, s = 0.06, z = 0 and different values of g = 0.0352, 0.0362, 0.0372.

0.5

reduced frequency

20

200

0 0.02

0.04

0.06

0.08

0.1

s=0.060 , z=0 , g=0.0352 s=0.063 , z=0 , g=0.0352

P2

100

P0 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

reduced frequency

Fig. 5. The plot of spectral density function S(Q, x) vs. reduced ~ ¼ ðx=xQ Þ for fixed values of e = 0.025, p = 1.0, frequency x t = 0.001, z = 0, g = 0.0352 and different values of electron– phonon interaction s = 0.060, 0.063.

phonon in the lattice, while P2b is due to the SDW gap excitation. It appears at a frequency ~ ’ 0:083, which is slightly greater than the pure x SDW gap 2g = 0.0704. Hence the SDW gap appearing in the spectral density function is renormalised to a slightly higher value in presence of electron–phonon interaction. The dependence of

G.C. Rout et al. / Physica C 420 (2005) 37–50

these two peaks on SDW order parameter g and the temperature t is further investigated in Figs. 6 and 7. The effect of increasing the SDW order parameter g on excitation peaks of Fig. 5 is shown in Fig. 6. When SDW order parameter increases effectively, its coupling to the phonon is suppressed. As a result, the peak P2a attributed to the local phonon distortion decreases in spectral 450 g=0.0352

P2a

400

g=0.0424 g=0.0430

350

g=0.0440

SDF

300 250

P2b

200 150 s

g

100 50 0 0.03

0.05

0.07

0.09

0.11

reduced frequency

Fig. 6. The plot of spectral density function S(Q, x) vs. reduced ~ ¼ ðx=xQ Þ for fixed values of e = 0.025, p = 1.0, frequency x t = 0.001, s = 0.063, z = 0 and different values of g = 0.0352, 0.0424, 0.0430, 0.0440.

45

weight and shifts in frequency towards the peak P2b. Under this condition, the spectral weight and the position of the peak P2b almost remains unaltered. On increasing the SDW order parameter to g = 0.043, the peak P2a merges with P2b, and with further increase of g, the peak gradually gets suppressed in intensity. The effect of increasing the temperature (t) on these peaks as shown in Fig. 7 is very similar to those of increasing SDW order g. In principle, with increasing temperature, the frequency of the peak should decrease since g also decreases, which is not accounted for in the present calculation. The superconducting gap excitation peak (P1) in Fig. 1 appears in presence of a weak electron– phonon coupling, s = 0.04 in absence of SDW phase. When the electron–phonon interaction is increased to a higher value s = 0.06 in the strong coupling range, the same peak in absence of SDW phase (g = 0) splits into two peaks with P1 ~ ’ 0:35 and P2 at x ~ ’ 0:25 as shown in Fig. at x 8. Of the two peaks, the one at P1 can be identified as the superconducting gap excitation peak (same as that of Fig. 1) appearing at the gap value of ~ ’ 0:2 2z ’ 0.35 while the peak P2 appearing at x is similar to the one appearing for strong interaction s = 0.063 in the SDW case in Fig. 5. However,

100

300

s=0.06, z=0.176, g=0.0352 s=0.06, z=0.176, g=0.000

P2a

P2

P2b 200

80

SDF

SDF

t=0.0010 t=0.0100 t=0.0125 t=0.0150

100

P2

60

(g = 0.0424)

p1

50

40 20 0 0.23

0.24

0.25

0.26

P0

0 0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.11

0.12

reduced frequency

Fig. 7. The plot of spectral density function S(Q, x) vs. reduced ~ ¼ ðx=xQ Þ for fixed values of e = 0.025, p = 1.0, frequency x s = 0.063, z = 0, g = 0.0424 and different values of t = 0.0010, 0.0100, 0.0125, 0.0150.

0

0

0.25

0.5

0.75

1

reduced frequency

Fig. 8. The plot of spectral density function S(Q, x) vs. reduced ~ ¼ ðx=xQ Þ for fixed values of e = 0.025, p = 1.0, frequency x t = 0.001, s = 0.06, z = 0.176 and different values of g = 0.0352 and 0.0.

G.C. Rout et al. / Physica C 420 (2005) 37–50 30 2.6

P1

P1+P2

2.1

20

P2

SDF

even in the coexistent SDW–SC case corresponding to the SDW order parameter g = 0.0352, the second peak (dotted curve shown in the inset of ~¼ Fig. 8) still coincides with the peak P2 at x 0:25. Its frequency being much larger than the SDW gap 2g (=0.0704), thus the peak P2 either has no bearing to the SDW order or as a consequence of the coexistence, there is an enhancement of the SDW state [30]. The interplay of this peak P2 with the SC peak P1 is further explored in Figs. 9–12. The effect of increasing the value of the SDW order parameter (g) on the peaks P1 and P2 when the electron–phonon interaction strength is kept fixed at s = 0.06 is depicted in Fig. 9. The increase of the SDW order parameter to the value g = 0.150 shifts the peak (P2) towards the higher frequencies ~ ¼ 0:300) and shifts the SC gap excitation peak (x (P1) towards the lower frequencies indicating that P2 is indeed the peak corresponding to the SDW gap. For a still higher value of g ’ 0.188, the two peaks merge to form a single peak with much diminished spectral weight. This shows that the presence of a dominant SDW order suppresses the SC order and accordingly its contribution to the SDF. In continuation to the results of Fig. 9, when the SDW order parameter g is further increased

1.6 0.34

0.39

10

g = 0.186 g = 0.188 g = 0.191

0 0.3

0.4

0.5

reduced frequency

Fig. 10. The plot of spectral density function S(Q, x) vs. ~ ¼ ðx=xQ Þ for fixed values of e = 0.025, reduced frequency x p = 1.0, t = 0.001, s = 0.06, z = 0.176 and different values of g = 0.186, 0.188, 0.191.

P2 80 z=0.156 z=0.176 z=0.196

P1

60

SDF

46

40

80

P0

20

P2

P1

P2

60

20

10

0

P1

0

0.2

0.4

0.6

0.8

1

SDF

reduced frequency 0 0.2

40

0.25

0.3

0.35

0.4

Fig. 11. The plot of spectral density function S(Q, x) vs. ~ ¼ ðx=xQ Þ for fixed values of e = 0.025, reduced frequency x p = 1.0, t = 0.001, s = 0.06, g = 0.0352 and different values of z = 0.156, 0.176, 0.196.

g=0.0352

p0

g=0.1500 g=0.1880

20

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

reduced frequency

Fig. 9. The plot of spectral density function S(Q, x) vs. reduced ~ ¼ ðx=xQ Þ for fixed values of e = 0.025, p = 1.0, frequency x t = 0.001, s = 0.06, z = 0.176 and different values of g = 0.0352, 0.1500, 0.1880.

beyond g = 0.188, an interesting feature is observed as shown in Fig. 10 i.e. for g = 0.191, the single peak formed due to the merging of the peaks P1 and P2 (dashed curve of Fig. 9), develops a shoulder which could be the peak P2 due to the SDW excitation, very close to the original peak P1 (i.e. the SC gap excitation peak) of much dimin-

G.C. Rout et al. / Physica C 420 (2005) 37–50

action suppresses the effect of SDW order. Hence the phonon modulated SDW gap excitation peak P2 shifts to lower frequencies.

100 75

P2

P1

50

P2

25

SDF

0 0.18

0.28

6. Conclusion

0.38

P1

50

s=0.058, z=0.176, g=0.0352 s=0.060 s=0.063 s=0.068 P0

0

47

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

reduced frequency

Fig. 12. The plot of spectral density function S(Q, x) vs. ~ ¼ ðx=xQ Þ for fixed values of e = 0.025, reduced frequency x p = 1.0, t = 0.001, z = 0.176, g = 0.0352 and different values of s = 0.058, 0.060, 0.063, 0.068.

ished spectral weight. These features when magnified in the inset of Fig. 10, clearly resemble the observed experimental peaks of some of the borocarbide systems [20,21]. The effect of varying the SC order parameter on the two excitation peaks is shown in Fig. 11. The increase of SC order (z) shifts both the excitation peaks P1 and P2 towards higher frequencies accompanied by a reduction in spectral weights. The increase in SC order means, the increase in Cooper pair binding, which is caused by the corresponding increase in electron–phonon interaction strength for phonon mediated superconductors. The increase of electron–phonon interaction also shifts the phonon modulated SDW excitation peak (P2) to higher energies as shown by Fig. 11. Thus the electron–phonon (e–p) interaction plays a vital role in producing superconducting excitation and the SDW excitation peaks in the SDF; the later being observable only in presence of strong electron–phonon interaction. The effect of electron–phonon coupling (s) on these two peaks P1 and P2 is shown in Fig. 12. As s increases, i.e., the e–p interaction strength gets enhanced, the Cooper pair binding gets enhanced and thereby the SC gap excitation peak P1 shifts to higher frequencies. The enhanced electron–phonon inter-

In conclusion, we summarise the main results of this paper. The phonon spectral density function for a system with coexistent SDW–SC state is evaluated from the Green
s function for the SDW zone folded (q = Q) phonon. The density response function and hence the phonon self-energy is calculated for the coexistent SDW–SC phase. These results are compared with the available experimental data on borocarbide and high-Tc superconductors, from neutron scattering and Raman scattering measurements. In principle the neutron and Raman scattering intensities are proportional to the phonon spectral density functions for finite wave vector (q) and for q = 0, and Q respectively. The q = Q zone centre mode is a consequence of the zone folding due to the SDW state present in the system. Inelastic light scattering proceeds by exciting only the zone centre (q = 0 and Q) modes because of the requirement of momentum conservation. 6.1. For borocarbide superconductors As discussed in Section 5, the phonon spectral density function (SDF) shows a peak P2 corresponding to the gap excitation (2g) mode of the SDW state only for strong e–p coupling with strength s = 0.060 (Fig. 5). When the strength of the coupling is increased to s = 0.063, the peak ~ ¼ 0:040 and P2b P2 splits into two peaks P2a at x ~ ¼ 0:083. The two peaks do not show up for at x smaller values of the coupling constant. In Fig. 6, in the presence of only the SDW state, the behaviour of the two peaks as a function of the SDW order parameter is depicted. The peak P2b which is due to the SDW gap excitation mode remains robust while the peak P2a which could be due to the lattice instability arising from strong e–p coupling and perfect nesting moves towards P2b with increasing value of the SDW order parameter (g) and merges with it. This could be

48

G.C. Rout et al. / Physica C 420 (2005) 37–50

an indication of the fact that with increasing g the lattice instability (an incipient CDW state) gets suppressed. Similar two peaks are observed in neutron scattering measurements on the borocarbide systems LuNi2B2C and YNi2B2C [20,21]. In these systems because of the strong electron–phonon interaction, phonons of an acoustic and an optic branches exhibit pronounced Kohn anomalies for wave vector close to the Fermi surface nesting wave vector Q which is incommensurate. The energies of the modes with wave vector close to Q decreases with decreasing temperature and the intensity is transferred from the higher to lower mode, an observation typical of the mode coupling behaviour [34,35]. At temperatures above Tc, these modes are so strongly coupled that the two peaks cannot be resolved. In the present calculation though the value of the electron–phonon coupling (s = 0.063) is strong in the normal SDW phase, it is smaller than the values used for the CDW superconductor (where s = 0.11–0.15) [26,31] while larger than that of the case of the two sublattice antiferromagnetic state (where s = 0.025) [36]. Besides, it is also assumed that there is perfect nesting and the nesting wave vector Q is commensurate. In the coexistent (SDW + SC) phase, the spectral density function shows two peaks i.e. P1 and P2 besides the normal zone folded SDW phonon peak P0. The peak P1 is clearly related to the superconducting gap excitation and the peak P2 is associated with the SDW order parameter. The SDW order parameter peak P2 is observable only in the presence of a very strong electron–phonon interaction; otherwise the peak P2 is suppressed even in the presence of a large value of the SDW order parameter. In consequence, we observe a sharp superconducting gap excitation peak P1 and a flat weak peak P2 very close to P1 as shown by the dashed curve in the inset of Fig. 9. It is to be noted that the opening of the superconducting gap does not appreciably affect the Fermi surface nesting in these compounds, as one would have expected [37]. The results of this model study clearly show that (see Figs. 9 and 10) the origin of the dramatic change in the phonon spectra is due to the onset of superconductivity in borocarbide systems as shown experimentally [20,21]. However, it should be pointed out that the calcu-

lations are carried out for a commensurate SDW wave vector (Q) with perfect nesting, while the systems on which the experimental data exists show incommensurate behaviour. Earlier, the opening of the superconducting gap is well understood and has been observed [35,37] in the superconducting materials Nb and Nb3Sn. A phonon mode with energy lower than the superconducting gap 2D cannot decay by breaking up into a Cooper pair; therefore its life time increases (line width narrowing) for T < Tc as compared to its life time above Tc. Further because of the singularity at x ’ 2D in the polarisability of the superconducting state, there is a shift in the phonon frequencies as well as a change in the Raman or neutron spectra as the temperature is decreased below Tc. Both the shift in phonon frequency and its line shape depend on the separation in energy between the phonon and the superconducting gap 2D. It is interesting to note that when the phonon energy is close to 2D, its line shape calculated in the present model is similar to that observed experimentally [20,21]. This type of anomalous behaviour is shown to exist only for phonons with wave vectors close to the nesting wave vector Q in confirmation with the experimental observation. Kee and Varma [38] have also found that the electronic polarisability for an extremum wave vector of the Fermi surface exhibits a pole for frequencies close to 2D. 6.2. For High-Tc superconductors For the CDW superconductors, the Raman spectrum exhibits the SC gap excitation at 2D, only if the phonon frequency is close to the SC gap. This was first shown by experiment [25] and also explained subsequently by theoretical work [39,40]. Analogous to the case of the CDW–SC state, even in the SDW–SC, SC gap excitation can be observed in Raman scattering in presence of a strong electron–phonon interaction as shown in Fig. 8. This state can be realised in the systems, where SC arises due to a phonon mediated interaction. The numerical analysis further shows, as g increases and becomes Pz, the phonon assisted SDW peak moves towards the SC peak and merges with it, resulting in a broad single peak

G.C. Rout et al. / Physica C 420 (2005) 37–50

with reduced intensity (see inset of Fig. 9). The Raman spectra of high-Tc superconductors do show such features. However, in high-Tc superconductors the mechanism of pairing is not due to phonon mediation. Besides it is now established that in these materials, the superconducting symmetry is not s-wave, but more like d-wave [41]. These materials are known to be antiferromagnets at low dopant concentrations, whereas superconductivity arises at relatively large dopant concentrations. If one assumes the antiferromagnetism to be an SDW state, even then the possibility of coexistence of the SDW and SC states needs to be established. It has been shown [29] that only for s-wave pairing symmetry of the SC state, there is possibility of coexistence of SC with SDW state for dopant concentrations x 6 0.5. On the other hand for pairing symmetries of either the extended s-wave type or the d-wave type, there is no coexistence of the two long range orders. They appear at different ranges of dopant concentrations. The present calculation has been carried out for the coexistent SDW–SC state, with s-wave SC pairing symmetry. In view of the above discussion, we examine the Raman scattering data of high-Tc superconductors from the point of view of the present calculation. There have been several attempts to observe SC peak in the Raman spectrum of the high-Tc superconductor YBa2Cu3O7
d. Assuming the high-Tc materials to be a BCS superconductors (2D/ kBTc = 3.52), the SC gap mode of the YBCO is expected to appear around 200 cm
1 as shown by the calculation of Behera and Mishra [42]. No such peak is observed experimentally [43]. In contrast to the expected sharp peak around the SC gap, the observed Raman spectra shows a broad peaked background at temperature below Tc, over which ride the sharp phonon peaks [44,45]. The appearance of the broad peaked background with almost vanishing intensity at low frequencies and a maximum around 470 cm
1, is again an indication of the presence of a gap [45]. However, the observed frequency of 470 cm
1 is much too high compared to the BCS predictions. This peak has been attributed to the electronic Raman scattering rather than the scattering by phonons. It has been argued that the observed peak at 470 cm
1 is the

49

collective mode of the coexistence SDW–SC phase appearing at a frequency of 2(g + z) as predicted by Behera and Bhattacharya [46].

Acknowledgments The authors (GCR and KCB) would like to gracefully acknowledge the research facilities offered by the Institute of Physics, Bhubaneswar, Orissa, India during their short stay. One of the authors (GCR) would like to thank the UGC, New Delhi for providing financial assistance vide letter No:PSO-015/02(ERO)Dt:22.11.2002.

References [1] [2] [3] [4] [5] [6] [7] [8]

[9]

[10]

[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

C. Mazumdar et al., Solid State Commun. 87 (1993) 413. L.C. Gupta, Physica B 223–224 (1996) 56. H. Eisaki et al., Phys. Rev. B 50 (1) (1994) 647. S.N. Behera, B.N. Panda, G.C. Rout, P. Entel, Int. J. Mod. Phys. B 15 (19–20) (2001) 2519. P.C. Canfield, P.L. Gammel, D.J. Bishop, Phys. Today (October) (1998). P. Dervenagas, J. Zarestky, C. Stassis, A.I. Goldman, P.C. Canfield, B.K. Cho, Physica B 212 (1995) 1. C. Stassis, A. Goldman, J. Alloys Compd. 250 (1997) 603. J. Zarestky, C. Stassis, A.I. Goldman, P.C. Canfield, P. Dervenagas, B.K. Cho, D.C. Johnston, Phys. Rev. B 51 (1995) 678. S.K. Sinha, J.W. Lynn, T.E. Grigereit, Z. Hossain, L.C. Gupta, R. Nagarajan, C. Godart, Phys. Rev. B 51 (1995) 681. A.I. Goldman, C. Stassis, P.C. Canfield, J. Zarestky, P. Dervenagas, B.K. Cho, D.C. Johnston, Phys. Rev. B 50 (1994) 9668. T. Vogt, A. Goldman, B. Sternlieb, C. Stassis, Phys. Rev. Lett. 75 (1995) 2628. P. Dervenagas, J. Zarestky, C. Stassis, A.I. Goldman, P.C. Canfield, B.K. Cho, Phys. Rev. B 53 (1996) 8506. C. Detlefs, A.I. Goldman, C. Stassis, P.C. Canfield, B.K. Cho, J.P. Hill, D. Gibbs, Phys. Rev. B 53 (1996) 6355. J.Y. Rhee, X. Wang, D.N. Harmon, Phys. Rev. B 51 (1995) 15585. W.E. Pickett, D.J. Singh, Phys. Rev. Lett. 72 (1994) 3702. L.F. Mattheiss, Phys. Rev. B 49 (1994) 13279. R. Coehoorn, Physica C 228 (1994) 5671. L.F. Mattheiss, T. Siegrist, R.J. Cava, Solid State Commun. 91 (1994) 587. K. Kawano, H. Yoshizawa, H. Takeya, K. Kadownki, Phys. Rev. Lett. 77 (1996) 4628. C. Stassis et al., Phys. Rev. B 55 (14) (1997) R8678. M. Bullock et al., Phys. Rev. B 57 (13) (1998) 7916.

50

G.C. Rout et al. / Physica C 420 (2005) 37–50

[22] D.C. Johnston, S.K. Sinha, A.J. Jacobson, J.M. Newsam, Physica C 153–155 (1988) 572. [23] M.R. Mcfarlane, H. Rosen, H. Seki, Solid State Commun. B 63 (1987) 831. [24] R.J. Schrieffer, X.-G. Wen, S.C. Zhang, Phys. Rev. Lett. 60 (1988) 944. [25] R. Sooryakumar, M.V. Klein, Phys. Rev. Lett. 45 (1980) 660. [26] C.A. Balseiro, L.M. Falicov, Phys. Rev. Lett. 45 (1980) 662. [27] R. Zeyher, G. Zwicknagl, Physica C 153–155 (1988) 1279. [28] J.R. Schrieffer, X.-G. Wen, S.C. Zhang, Phys. Rev. B 39 (1989) 11663. [29] Harnath Ghosh, S. Sil, S.N. Behera, Physica C 316 (1999) 34. [30] K.C. Bishoyi, G.C. Rout, S.N. Behera, Indian J. Phys. 78 (2004) 809. [31] B. Pradhan, G.C. Rout, S.N. Behera, Indian J. Phys. 77A (6) (2003) 581. [32] J.C. Swhart, Phys. Rev. 116 (1959) 45. [33] P. Morel, P.W. Anderson, Phys. Rev. 125 (1962) 1263. [34] J.D. Axe, G. Shirane, Phys. Rev. Lett. 30 (1973) 214.

[35] S.M. Shapiro, G. Shirane, J.D. Axe, Phys. Rev. B 12 (1975) 4899. [36] K.C. Bishoyi, G.C. Rout, S.N. Behera, Indian J. Phys. 77A (6) (2003) 593. [37] T.V. Ramakrishnan, C.M. Varma, Phys. Rev. B 24 (1981) 137. [38] H.Y. Kee, C.M. Varma, Phys. Rev. Lett. 79 (1997) 4250. [39] P.B. Littlewood, C.M. Varma, Phys. Rev. Lett. 47 (1981) 811. [40] S.N. Behera, S.G. Mishra, Phys. Rev. B 31 (1985) 2773. [41] J. Ruralds, C.T. Rieck, S. Tewar, J. Thoma, A. Virosztek, Phys. Rev. B 51 (1995) 3797. [42] S.N. Behera, S.G. Mishra, Proc. Solid State Phys. Sympos. (India) 30C (1987) 243. [43] Z. Iqbal, S.W. Steinhauser, A. Bose, N. Cipollini, H. Eckhardt, Phys. Rev. B 36 (1987) 2283. [44] K.B. Lyons, S.H. Liou, M. Hong, H.S. Chen, J. Kwo, T.J. Negran, Phys. Rev. B 36 (1987) 5592. [45] S.L. Cooper, M.V. Klein, B.C. Bazol, J.P. Rice, D.M. Ginsberg, Phys. Rev. B 37 (1988) 5920. [46] S.N. Behera, S. Bhattacharya, in: Proc. Third Asia Pacific Physics Conference (Hong Kong), Phase Transit. 19 (1989) 15.